Séminaire SPACE Tours
Connecting analytic and asymptotic behaviour through multivariate diagonalsStephen Melczer (Univ. of Waterloo & ENS Lyon)
Friday 06 May 2016 11:00 - Tours - Salle 1180 (Bât E2)
Résumé :
Recent work in the study of analytic combinatorics in several variables has shown how to derive asymptotics for the coefficients of certain families of D-finite functions by representing them as diagonals of multivariate rational functions. In this talk we look at applications of this theory to the enumeration of two dimensional lattice paths in restricted regions: the classical "kernel method" allows one to represent the generating functions of many combinatorial classes arising in this context as diagonals, whose analytic behaviour can then be studied. In particular, we examine a close link between combinatorial properties of the lattice path models (such as their "drift"), the singularities of the associated multivariate rational functions, and the asymptotics of their counting sequences. This allows us to prove conjectured asymptotics of Bostan and Kauers (2009) for walks in two dimensions restricted to the positive quadrant, and newer conjectures of Bostan, Chyzak, van Hoeij, Kauers, and Pech (2015) on the number of walks returning to each bounding axis and the origin. This is joint work with Mark Wilson.
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